Iran's supreme leader Ayatollah Ali Khamenei on Friday insisted that the Islamic republic would 'not back down' in the face of protests after the biggest rallies yet in an almost two week movement sparked by anger over the rising cost of living.

Chanting slogans including 'death to the dictator' and setting fire to official buildings, crowds of people opposed to the clerical establishment marched through major cities late Thursday.

Internet monitor Netblocks said authorities had imposed a total connectivity blackout late Thursday and added early Friday that the country has 'now been offline for 12 hours... in an attempt to suppress sweeping protests'.

The demonstrations represent one of the biggest challenges yet to the Islamic republic in its over four-and-a half decades of existence, with protesters openly calling for an end to its theocratic rule.

But Khamenei struck a defiant tone in his first comments on the protests that have been escalating since January 3, calling the demonstrators 'vandals' and 'saboteurs', in a speech broadcast on state TV.

Khamenei said 'arrogant' Donald Trump's hands 'are stained with the blood of more than a thousand Iranians', in apparent reference to Israel's June war against the Islamic republic which the US supported and joined with strikes of its own.

He predicted the US leader would be 'overthrown' like the imperial dynasty that ruled Iran up to the 1979 revolution.

'Last night in Tehran, a bunch of vandals came and destroyed a building that belongs to them to please the US president,' he said in an address to supporters, as men and women in the audience chanted the mantra of 'death to America'.

'Everyone knows the Islamic republic came to power with the blood of hundreds of thousands of honourable people, it will not back down in the face of saboteurs,' he added.

Trump said late on Thursday that 'enthusiasm to overturn that regime is incredible' and warned that if the Iranian authorities responded by killing protesters, 'we're going to hit them very hard. We're ready to do it.'

He also claimed the Ayatollah was 'looking to go someplace' amid reports he may flee to Russia.  

The protests also represented the first test of whether the Iranian public could be swayed by Crown Prince Reza Pahlavi, whose fatally ill father fled Iran just before the country's 1979 Islamic Revolution. Pahlavi, who called for the protests Thursday night, similarly has called for demonstrations at 8pm Friday.